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Communications in Mathematical Physics

, Volume 303, Issue 3, pp 785–824 | Cite as

Concentration of Measure for Quantum States with a Fixed Expectation Value

  • Markus P. Müller
  • David Gross
  • Jens Eisert
Article

Abstract

Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors \({|\psi\rangle}\) that have a fixed expectation value \({\langle\psi|H|\psi\rangle=E}\) with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.

Keywords

Quantum State Isoperimetric Inequality Reduce Density Matrix Gibbs State Pure Quantum State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Markus P. Müller
    • 1
    • 2
  • David Gross
    • 3
    • 4
  • Jens Eisert
    • 2
    • 5
  1. 1.Institute of MathematicsTechnical University of BerlinBerlinGermany
  2. 2.Institute of Physics and AstronomyUniversity of PotsdamPotsdamGermany
  3. 3.Institute for Theoretical PhysicsLeibniz University HannoverHannoverGermany
  4. 4.Institute for Theoretical PhysicsETH ZürichZürichSwitzerland
  5. 5.Institute for Advanced Study BerlinBerlinGermany

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